Truth value? ∀x∃y (x=y²)

Section 1.4
Determine the truth value and state why.
28b. ∀x∃y: x=y²

hmmm?

If you have worked with me, then you know what I'm going to say. Let's rewrite this to help our visual memory. Since most of us are accustomed to seeing the "y" on the left-hand side of the equations, let's write this:

from: ∀x∃y: x=y²

to: ∀x∃y: y²=x

Solution:

y²=x

√y² = √x
y = √x.

For this to be true, then x must be only non-negative integers. Therefore, our original statement is false.

Show that ∃n∃m (n² + m² = 6) is False

Section 1.4
27f. Prove that ∃n∃m (n² + m² = 6) is False.

The only possible pairs of integers whose sum is 6 are 1+5, 2+4, and 3+3. Since none of 2, 3, or 5 are squares of integers, then clearly the pairs of integers whose sum is 6 are not solutions that satisfy ∃n∃m (n² + m² = 6).